# M\"obius Quantum Walk

## TL;DR

This paper introduces M"obius Quantum Walk, a novel quantum walk model with a tunable parameter that allows for uniform distribution on cycles, unlike traditional quantum walks.

## Contribution

It presents a new quantum walk model with an extra Hilbert space, enabling control over distribution uniformity through a parameter, expanding quantum walk theory.

## Key findings

- Tuning the M"obius factor $oldsymbol{}$ achieves uniform distribution for any cycle size.
- Derived explicit formulas for non-uniform limiting distributions.
- Analytically investigated the effects of the M"obius factor on quantum walk behavior.

## Abstract

By adding an extra Hilbert space to Hadamard Quantum Walk on Cycles (QWC), we presented a new type of QWCs called M\"obius Quantum Walk (MQW). The new space configuration enables the particle to rotate around the axis of movement. We defined factor $\alpha$ as the M\"obius factor which is number of rotations per cycle. So by $\alpha=0$ we have normal QWC, while $\alpha \neq 0$ defines new type of QWC (namely M\"obius Quantum Walk). Specially $\alpha = \frac{1}{2}$ defines a structure similar to M\"obius strip. We analytically investigated this new type of QW and found that by tuning the parameter $\alpha$ we can reach uniform distribution for any number of nodes, while it is impossible for QWC. The effects of $\alpha$ on limiting distribution have been investigated and an explicit formula for non-uniform cases has been derived as well.